Lecture 03

Author

Bill Perry

Lecture 2: Review

We covered:
- data wrangling and types of variable names
- meta data
- project design
- summary statistics
- graphing the mean and standard error graphs
- pipes and %>% or |> and how to group_by

Our last graph

Lecture 3: How to deal with data wrangling

Introduction to probability distributions

What is a frequency distribution?
What is a probability distribution?
Distributions for variables and for statistics

Estimation

Populations and samples
Parameters and statistics

we are going to use some sculpin data that is real!

Wikipedia

Lecture 3: Frequency distributions

Example data we will use will be a combination of data from Toolik Alaska LTER

source

We will specifically look at fishes like

Lake Trout

source

Grayling

Slimy Sculpin

source

Lecture 3: Frequency distributions

Data - has been cleaned in terms of lake names and species names
Slimy Sculpin - Toolik Lake

# A tibble: 1 × 4
   mean    sd    se count
  <dbl> <dbl> <dbl> <int>
1  51.7  12.0 0.834   208

Lecture 3: Frequency Distributions

What is a frequency distribution?

Display of number of observations in certain intervals
e.g., the number of sculpin per interval in Toolik Lake
as a table like below or histogram

# A tibble: 28 × 2
   length_bin     n
   <fct>      <int>
 1 [11,13]        4
 2 (19,21]        1
 3 (23,25]        1
 4 (27,29]        2
 5 (29,31]        2
 6 (31,33]        1
 7 (33,35]        4
 8 (35,37]        3
 9 (37,39]        7
10 (39,41]        9
# ℹ 18 more rows

Lecture 3: Frequency Distributions

The alternative is to use a histogram

the y axis is the count
the x axis is the bin range
each bin 0 - 5 and 5 - 10 and 10 - 15 or as you choose
in ggplot the code looks like

dataframe %>% ggplot(aes(thing_to_count))+
  geom_histogram( 
    binwidth = increments_to_work_with
    )

Lecture 3: Frequency Distributions

What happens as sample size changes…

Low sample number - 15

High sample number - 70

Frequency distribution takes on “bell-shape”…

Lecture 3: Probability distributions

Can we make assumption about distribution of random variable weight in population?

Probability distribution:

theoretical frequency distribution in population

Lecture 3: Probability distributions

For continuous random var: probability density function (PDF)
PDF: mathematical expression of probabilities associated with getting certain values of random variable
Area under curve = 1
i.e., probability of lenght between 10 and 80 = 1

Lecture 3: Probability distributions

Now we could look at a lot of different ranges of lengths - probability of the lenght larger than the mean - probability of the lenght larger than 70 mm - probabilioty of the lenght between two numbers

Lecture 3: Probability distributions

Usually need to know probability distribution of random variables in statistical analyses
Can define many distributions; some do reasonable job especially whit continuous varaibles
Different distributions for continuous, discrete variables like a single die

asdfasdf

Lecture 3: Probability distributions

Normal (Gaussian): symmetrical, bell-shaped

Defined in terms of mean and variance (μ, 𝜎2)
SND (z-distribution) has mean μ=0 , 𝜎2 =1

\[f(y) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(y - \mu)^2}{2\sigma^2}}\]

Lecture 3: Probability distributions

Lognormal: right-skewed distribution
Logarithm of random variable is normally distributed
Common in biology.
Why would this occur or be common in biology?

Lecture 3: Probability distributions

Binomial (multinomial):

probability of event that have two outcomes (heads/ tails, dead/alive)
Defined in terms of “successes” out of set number of trials

In large number of trials: approx. normal distribution

Lecture 3: Probability distributions

Poisson: occurrences of (rare) event in time/space

E.g., number of
- Taraxacum officinale - common dandelion in quadrat
- copepod eaten per minute
- cells in field of view
Measures Probability(y= certain integer value)
- defined in terms of μ or mean
- Right-skewed at small μ
- more symmetrical at higher μ

Lecture 3: Data gathering - managing

Also have distributions of test statistics

Test statistics:

summary values calculated from data used to test hypotheses
is your result due to chance?

Different test statistics:

different, well-defined distributions
allows estimation of probabilities associated with results
Examples:
- z-distribution, student’s t-distribution, χ2-distribution, F-distribution

Lecture 3: Samples and populations

asdfadsf

asdfasdf

Lecture 3: Samples and populations

asdfadsf

asdfasdf

Lecture 3: Samples and populations

asdfadsf

asdfasdf

Lecture 3: Samples and populations

Inferential statistics:

inference from samples to populations

Statistical population:

All possible observations of interest
Normally: populations too large to census

Populations are defined in time + space

Examples of statistical populations from you research area?

Lecture 3: Samples and populations

Key characteristic of sample is

size (n observations; n = sample size)

Characteristics of population - called parameters

Parameters - Greek letters

Characteristics of samples - statistical estimates of parameters

statistics= Latin letters

Random sampling crucial for

sample -> population

inference statistics -> parameters

Lecture 3: Parameters and statistics

Two main kinds of summary statistics: - center and spread

Center: - Mean (µ, ȳ): sum of sampled values divided by n - Mode: the most common number in dataset - Median: middle measurement of data; = mean for normal distributions

\(\mu = \frac{\sum\limits_{i=1}^{n} Y_i}{n}\)

Formula for n odd \(Y_{(n+1)/2} \text{ if } n \text{ odd}\)

Formula for n even \(\frac{Y_{n/2} + Y_{(n/2)+1}}{ 2} \text{ if } n \text{ even}\)

Lecture 3: Parameters and statistics

Spread

Range: from highest and lowest observation
Variance (σ2, s2): sum of squared differences of observations from mean, divided by n-1

E.g., fish lengths = 20, 30, 35, 24, 36 g

# A tibble: 1 × 1
   mean
  <dbl>
1    29

\(s^2 = \sum_{i=1}^{n} \frac{(y_i - \bar{y})^2}{n-1}\)

Lecture 3: Parameters and statistics

Spread

(20 -29)^2+ (30 -29)^2 + (35 -29)^2 + (24 -29)^2 + (36 -29)^2 = 57,104

192 / (5-1) = 48 mm^2 Problem: weird units!

# A tibble: 1 × 2
   mean variance
  <dbl>    <dbl>
1    29       48

Lecture 3: Parameters and statistics

Spread

Standard Deviation(σ, s): square root of variance.
- In same units as observations
- In example: √48 = 6.9 mm
Coefficient of variation: SD as % of mean.
- Useful for comparing spread in samples with different means
- In example: (6.9/29)*100= 23.8 %

\(\sqrt{\sum_{i=1}^{n} \frac{(y_i - \bar{y})^2}{n-1}}\)

\(\frac{S}{\bar{Y}} X 100\)

Lecture 3: Estimation

Problem: - don’t know the values of parameters

Goal: - estimate parameters from empirical data (samples)

3 general methods of parameter estimation: - Maximum Likelihood Estimation (MLE) - Ordinary Least Squares (OLS) - Resampling techniques

MLE general method to estimate parameters in a way that maximizes the likelihood of the observed data given the parameter values.
aims to find the parameter values that make the observed data most probable under the assumed statistical model.
OLS specific method to estimate parameters of a linear regression model.
minimizes the sum of the squared differences between observed and predicted values

Other Formats

Lecture 2: Review

Lecture 3: How to deal with data wrangling

Introduction to probability distributions

Estimation

Lecture 3: Frequency distributions

Lecture 3: Frequency distributions

Lecture 3: Frequency Distributions

Lecture 3: Frequency Distributions

Lecture 3: Frequency Distributions

Lecture 3: Probability distributions

Lecture 3: Probability distributions

Lecture 3: Probability distributions

Lecture 3: Probability distributions

Lecture 3: Probability distributions

Lecture 3: Probability distributions

Lecture 3: Probability distributions

Binomial (multinomial):

In large number of trials: approx. normal distribution

Lecture 3: Probability distributions

Poisson: occurrences of (rare) event in time/space

Lecture 3: Data gathering - managing

Also have distributions of test statistics

Test statistics:

Different test statistics:

Lecture 3: Samples and populations

Lecture 3: Samples and populations

Lecture 3: Samples and populations

Lecture 3: Samples and populations

Inferential statistics:

Statistical population:

Populations are defined in time + space

Examples of statistical populations from you research area?

Lecture 3: Samples and populations

Key characteristic of sample is

Characteristics of population - called parameters

Characteristics of samples - statistical estimates of parameters

Lecture 3: Parameters and statistics

\(\mu = \frac{\sum\limits_{i=1}^{n} Y_i}{n}\)

Formula for n odd \(Y_{(n+1)/2} \text{ if } n \text{ odd}\)

Formula for n even \(\frac{Y_{n/2} + Y_{(n/2)+1}}{ 2} \text{ if } n \text{ even}\)

Lecture 3: Parameters and statistics

Lecture 3: Parameters and statistics

Spread

\(s^2 = \sum_{i=1}^{n} \frac{(y_i - \bar{y})^2}{n-1}\)

Lecture 3: Parameters and statistics

Lecture 3: Parameters and statistics

Spread

\(\sqrt{\sum_{i=1}^{n} \frac{(y_i - \bar{y})^2}{n-1}}\)

\(\frac{S}{\bar{Y}} X 100\)

Lecture 3: Estimation

Lecture 3: Estimation